Functional analysis. Vol. 1: A gentle introduction. (English) Zbl 1106.46001

Ha, Dzung Minh

Ithaca, NY: Matrix Editions (ISBN 0-9715766-1-0/hbk). xvi, 640 p. (2006).

This book is an introduction to basics of functional analysis at the undergraduate level. The prerequisites are elementary linear algebra and first-year calculus.
In the Preface, the author writes: “Textbooks in functional analysis (or more generally, in mathematics) are often unnecessarily demanding – written in a concise manner with few examples and motivations. \([\dots]\) I chose to write a textbook that I would like to have studied from as a student – one that is mathematically rigorous but leisurely, with lots of motivations and examples.”
The book discusses standard topics such as metric spaces and normed linear spaces and operators on them with rudiments of topological spaces and topological vector spaces, Banach’s fixed point theorem, compactness, results centering around the Baire category theorem, integral operators, inner product spaces and Hilbert spaces. It also includes special topics as the theorems of Korovkin and Bernstein, the Stone-Weierstrass theorem, the Baire-Osgood theorem, Gram determinants, the Müntz theorem and some basic results in the theory of differential equations. The Hahn-Banach theorem is not discussed. The spaces \(L_p[a,b]\) are not considered and measure theory is avoided. But notice that “two more volumes are planned”.
The author writes: “Proofs are developed in detail, with all steps justified. Almost all previously proven results used within proofs and solutions to examples are explicitly cited, and referred to by number. This eliminates unnecessary time and frustration spent figuring out exactly which results were used and where they were first proved.” On the other hand, one also could get frustrated, for instance, by the proof of the sequential continuity of distance (on page 57 of the book) which refers to three numbered facts, which in turn refer to four more numbered results.
The arguments in the book are detailed, but at the same time they often become tiresome and lengthy. I have the impression that, in mathematics, there are more students who do not like this kind of proofs but appreciate substantial and effective proofs which expose ideas behind them. The book also contains many elegant proofs.
The book contains about 250 examples and counter-examples. It also contains about 800 exercises. They complement the material in each of the 46 sections of the six chapters of the book. A remarkable feature is an appendix of 120 pages containing the complete solutions to all odd-numbered exercises.
Concerning terminology, it should be pointed out that instead of the common “first category” and “second category” (sets) the terms “thin” and “fat” are used in the book (recall that “thin” in Banach space theory has a different meaning: the set can be written as a countable increasing union of non-norming subsets).
The book is carefully written and its index is well arranged and comprehensive. There are just some typos in names: it should be Müntz (not Muntz), von Neumann (not Neumann), Runge (not Runga), Schröder (not Schroder).
The book will be useful to students from a wide variety of backgrounds, including graduate students in physics and engineering, who want to get a broad and slow introduction to the subject, together with applications.

Reviewer: Eve Oja (Tartu)

MSC:

46-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
47-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
40-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to sequences, series, summability
45-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral equations

Keywords:

metric spaces; normed spaces; Hilbert spaces; Baire category theorem; Stone-Weierstrass theorem; Müntz theorem